The Chemistry Maths Book, Second Edition

602 Chapter 21Probability and statistics

Table 21.4

Nn(H) f(H)

256 118 0.461

4040 2068 0.512

12000 6062 0.505

24000 11942 0.498

The results are not surprising because we expectheads (or tails) to come up about half

the time; the two outcomes are said to be ‘equally probable’. This experiment is an

example of a random experimentin which the outcomes depend entirely on ‘chance’

(the problem of what is meant by the words ‘random’, ‘chance’, and ‘probability’

has generated its own vast literature). Experience shows that random experiments

exhibit statistical regularity; that is, the relative frequency of a particular outcome

in a long sequence of trials remains about the same when several such sequences

are performed. In our example, and, for tails,. The

theoretical value of the relative frequency of an outcome, the value we expect, is the

probabilityof the outcome;.

Probability distributions

A set of possible outcomes (or events) {x

1

, x

2

, =, x

k

} with probabilities P(x

1

),

P(x

2

), =, P(x

k

)is called a probability distribution. Probability distributions are

the theoretical models (of populations) with which frequency distributions (of

samples) are compared for the analysis of experimental data. The total probability,

the probability that there be an outcome, is unity (for certainty), so that

(21.10)

The two most important properties of a probability distribution are its mean and

variance (or standard deviation).

The mean or expectation valueof xis

(21.11)

This is the sum in (21.3) with the relative frequenciesn

i

2 Nreplaced by probabilities.

The symbol μis that used most frequently, in statistics, for the mean of the population,

withEreserved for a sample of the population. Other symbols, that emphasize that

μis the expected value, areE(x)and〈x〉. Iff(x)is a function of x, the mean or

expectation value offis defined as

(21.12)

Ef f fxPx

i

k

ii

()=〈 〉= ()()

=

∑

1

μ=

=

∑

i

k

ii

xPx

1

()

i

k

i

Px

=

∑

=

1

() 1

PP() ()HT==

1

2

ff()=− 1 ( )TH≈

1

2

f()H ≈

1

2